cross

Cross product

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Syntax

C = cross(A,B)

C = cross(A,B,dim)

Description

example

C = cross(A,B) returns the cross product of A and B.

If A and B are vectors, then they must have a length of 3.
If A and B are matrices or multidimensional arrays, then they must have the same size. In this case, the cross function treats A and B as collections of three-element vectors. The function calculates the cross product of corresponding vectors along the first array dimension whose size equals 3.

example

C = cross(A,B,dim) evaluates the cross product of arrays A and B along dimension, dim. A and B must have the same size, and both size(A,dim) and size(B,dim) must be 3. The dim input is a positive integer scalar.

Examples

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Cross Product of Vectors

Open Live Script

Create two 3-D vectors.

A = [4 -2 1];
B = [1 -1 3];

Find the cross product of A and B. The result, C, is a vector that is perpendicular to both A and B.

C = cross(A,B)

C = 1×3

    -5   -11    -2

Use dot products to verify that C is perpendicular to A and B.

dot(C,A)==0 & dot(C,B)==0

ans = logical
   1

The result is logical 1 (true).

Cross Product of Matrices

Open Live Script

Create two matrices containing random integers.

A = randi(15,3,5)

A = 3×5

    13    14     5    15    15
    14    10     9     3     8
     2     2    15    15    13

B = randi(25,3,5)

B = 3×5

     4    20     1    17    10
    11    24    22    19    17
    23    17    24    19     5

Find the cross product of A and B.

C = cross(A,B)

C = 3×5

   300   122  -114  -228  -181
  -291  -198  -105   -30    55
    87   136   101   234   175

The result, C, contains five independent cross products between the columns of A and B. For example, C(:,1) is equal to the cross product of A(:,1) with B(:,1).

Cross Product of Multidimensional Arrays

Open Live Script

Create two 3-by-3-by-3 multidimensional arrays of random integers.

A = randi(10,3,3,3);
B = randi(25,3,3,3);

Find the cross product of A and B, treating the rows as vectors.

C = cross(A,B,2)

C = 
C(:,:,1) =

   -34    12    62
    15    72  -109
   -49     8     9


C(:,:,2) =

   198  -164  -170
    45     0   -18
   -55   190  -116


C(:,:,3) =

  -109   -45   131
     1   -74    82
    -6   101  -121

The result is a collection of row vectors. For example, C(1,:,1) is equal to the cross product of A(1,:,1) with B(1,:,1).

Find the cross product of A and B along the third dimension (dim = 3).

D = cross(A,B,3)

D = 
D(:,:,1) =

   -14   179  -106
   -56    -4   -75
     2   -37    10


D(:,:,2) =

   -37  -162   -37
    50  -124   -78
     1    63   118


D(:,:,3) =

    62  -170    56
    46    72   105
    -2   -53  -160

The result is a collection of vectors oriented in the third dimension. For example, D(1,1,:) is equal to the cross product of A(1,1,:) with B(1,1,:).

Input Arguments

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`A,B` — Input arrays
numeric arrays

Input arrays, specified as numeric arrays.

Data Types: single | double
Complex Number Support: Yes

`dim` — Dimension to operate along
positive integer scalar

Dimension to operate along, specified as a positive integer scalar. The size of dimension dim must be 3. If no value is specified, the default is the first array dimension whose size equals 3.

Consider two 2-D input arrays, A and B:

cross(A,B,1) treats the columns of A and B as vectors and returns the cross products of corresponding columns.
cross(A,B,2) treats the rows of A and B as vectors and returns the cross products of corresponding rows.

cross returns an error if dim is greater than ndims(A).

More About

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Cross Product

The cross product between two 3-D vectors produces a new vector that is perpendicular to both.

Consider the two vectors

$\begin{array}{l} A = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}, \\ B = b_{1} \hat{i} + b_{2} \hat{j} + b_{3} \hat{k} . \end{array}$

In terms of a matrix determinant involving the basis vectors $\hat{i}$ , $\hat{j}$ , and $\hat{k}$ , the cross product of A and B is

$\begin{array}{l} C = A \times B = | \begin{matrix} \begin{matrix} {\hat{i}}_{} & {\hat{j}}_{} \end{matrix} & {\hat{k}}_{} \\ \begin{matrix} \begin{matrix} a_{1} \\ b_{1} \end{matrix} & \begin{matrix} a_{2} \\ b_{2} \end{matrix} \end{matrix} & \begin{matrix} a_{3} \\ b_{3} \end{matrix} \end{matrix} | \\ = (a_{2} b_{3} - a_{3} b_{2}) \hat{i} + (a_{3} b_{1} - a_{1} b_{3}) \hat{j} + (a_{1} b_{2} - a_{2} b_{1}) \hat{k} . \end{array}$

Geometrically, $A \times B$ is perpendicular to both A and B. The magnitude of the cross product, $‖ A \times B ‖$ , is equal to the area of the parallelogram formed using A and B as sides. This area is related to the magnitudes of A and B as well as the angle between the vectors by

$‖ A \times B ‖ = ‖ A ‖ ‖ B ‖ \sin α .$

Thus, if A and B are parallel, then the cross product is zero.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

If supplied, dim must be a constant.
See Variable-Sizing Restrictions for Code Generation of Toolbox Functions (MATLAB Coder).
Code generation does not support sparse matrix inputs for this function.

GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Usage notes and limitations:

If supplied, dim must be a constant.
Code generation does not support sparse matrix inputs for this function.

GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays
Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Documentation

cross

Syntax

Description

Examples

Cross Product of Vectors

Cross Product of Matrices

Cross Product of Multidimensional Arrays

Input Arguments

`A,B` — Input arrays
numeric arrays

`dim` — Dimension to operate along
positive integer scalar

More About

Cross Product

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Distributed Arrays
Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

See Also

MATLAB Documentation

Support

Documentation

cross

Syntax

Description

Examples

Cross Product of Vectors

Cross Product of Matrices

Cross Product of Multidimensional Arrays

Input Arguments

A,B — Input arrays numeric arrays

dim — Dimension to operate along positive integer scalar

More About

Cross Product

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™.

GPU Code Generation Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

GPU Arrays Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Distributed Arrays Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

See Also

MATLAB Documentation

Support

`A,B` — Input arrays
numeric arrays

`dim` — Dimension to operate along
positive integer scalar

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Distributed Arrays
Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.